In March 2022 I was contacted by a person named Glagolj who has mathematically described what I’ve been calling “wheel-within-a-wheel” designs made with Wild Gears.

He prefers to call it “second order roulette” as it involves rolling one curve on top of another curve. Fair enough. For me, a mystery. For Glagolj, a mystery to be figured out and encapsulated in a parametric equation.

He wanted to use the patterns I’d drawn by hand and compare them side by side with the renderings generated by his javascript plotter. That’s the ultimate check on the formula. As you can see here, it works.

The plotter can be found here. It allows you to plug in the four gear numbers of a second order roulette, and generate the pattern they should give.

Glagolj’s article on the formula is here on Github.

He prepared a couple of slideshows with some of the interesting patterns he found. These are all computer-generated, so he can use any gear he wants. Of course, working with physical gears we have to work with the ones we have. But the numbers are in the upper-left if you want to try drawing them yourself.

Here is the second slideshow by Glagolj:

Several people have suggested I try third-order roulette patterns, i.e. a wheel-within-a-wheel-within-a-wheel. Well, Glagolj is ahead of me there. Now I have to try to see if I can duplicate his computer-generated results.

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Inside a 120-tooth ring, I used a gear with 96 teeth, which has an off-center hole with 40 teeth. (I don’t remember which set this came from.)

Then I drew the pattern six times with six different small gears inside the 40 ring to see the patterns.

Looking at the math of it

Each time I got a 5-lobed pattern. That’s the basic geometry of the 120-96 gear combination, 5 loops or points, as you can see in the table here.

But there’s a lot of variation in how that 5-lobed pattern is expressed, according to which small gear is used.

Small gears used are: 32 (navy blue ink), 26 (pink). 34 (purple), 18 (green), 30 (red), 16 (turquoise).

Using the same table, looking at the number of points each gear would give in a 40-tooth ring, we see that:

• Gears 18, 26 and 34 all give 20 points
• Gears 32 and 16 give 5 points
• Gear 30 gives 4 points.

Now the differences in pattern density make sense. The three 20 point patterns (pink, purple and green) look different depending on the physical size of the gear.

The other three looser patterns make more sense if you try to count the points or loops within each lobe. Some of the points are hidden in the inner part of the pattern, so they’re hard to make out. But they’re there. You may get an idea from watching the video.

So the overall geometry of a wheel-within-a-wheel pattern depends on the larger gear/ring pair, and the density depends on the inner, smaller gear/ring pair.

Here’s the video, filmed on a sunny summer day in direct sunlight. The glass-topped patio table gives a nice flat surface for this work.

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Conclusion: Spirograph is fun, but it’s a toy. Wild Gears is more expensive, and somewhat more difficult to use, but it’s a more serious drawing tool.

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Thanks for the great ideas! I’ve gone a more rudimentary, unwieldy, and not-so-artistic route but am mostly satisfied with it, though it’s a bit unwieldy.

I’ve used one of the cardboard boxes the Wild Gears ship in and used putty to adhere the gears to the inside of the box, arranged by size and labeled. I can fit the Full, Compact, and Strange Shapes gear sets on the large panel of the box and one of the two side flaps (except for the large gear set frames).

I like to be able to see everything at a glance, particularly when creating designs using a wheel within a wheel. That way I can see the size of the gear cutouts within the larger wheels. Keep the good ideas coming!

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The post Another Wild Gears Storage Idea appeared first on SpiroGraphicArt.

This is my response for how to use up the little ‘not much use for’ acrylic left over bits when unpacking Wild Gears.

it’s my first rain stick and I am quite happy with it. The acrylic pieces work really well, much lighter than seeds/pebbles.

I will make another one and will use more internal sticks and hopefully the acrylic will fall much slower.

A rainstick is an Australian Aboriginal percussion instrument that sounds like rain when you turn it one way or the other. Traditionally, it’s made from a dried cactus stick, with the thorns removed, turned around and pounded into the stick. The seeds, beans or pebbles inside knock against the thorns as they fall, making a tinkling sound.

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This is getting very deep into geeky gears country, but I bought a small set of metric nylon washers on Amazon. They were only a couple dollars so I was willing to experiment.

I took two slight different sized washers and glued them together with the holes aligned. This allowed use of a top washer with a hole sized so it wouldn’t ride up past the shaft of the pen tip and a bottom washer big enough to not drop into any of the holes on the gear. I put the washer pair over the gear hole I want to use and insert the pen through it. This traps the gear between the paper and the body of the pen, keeping the gear from riding up and popping out of the ring. It’s not the most elegant approach but it is reasonably effective for those cases where the gear keeps trying to pop out.

This is the kit of nylon washers I got on Amazon. None of these are a perfect fit as delivered. The ones in the lower right compartment have the right sized holes to not go further up than the lowest part of the body of the pen. Unfortunately, they’re so small they’ll drop into the medium sized gear holes. The other problem was none of them are thick enough to block the gear from riding up. My solution was to mix washers of two different sizes.

The gear on the right is the smallest one in the previous picture. The one on the left is two sizes larger. The hole on that one is too big and allows the washer to ride up too high on the pen. On the plus side, the larger one will not drop into the medium sized gear holes. So I glued two of these washers together.

This is an extreme close-up to show the detail. I used Super Glue. It worked but was probably not the best choice. It took forever to set up and it was difficult keeping the washers centered and pressed together while it set. It was definitely not a ultra fast cure like you used to see on the commercials. I used a nail to keep the washers centered as the glue dried and was afraid I’d end up with the washers glued to the nail. Were I to make a bunch of these I’d have to come up with a better approach, but for doing one or two it was adequate. I did run the mating surfaces over sandpaper to rough them up prior to gluing. The nylon is very slick and there’s almost nothing for the glue to hold on to unless you scratch it up first.

Here we have them kind of in action. Obviously not actually drawing anything here. Note how the washers keep the gear trapped against the paper and prevent it from jumping out of the ring. In this photo I think the tip of the pen is extended a bit further down than it would be if there were paper underneath. On paper there’s a bit of a gap between the body of the pen and the washers, but not so much that the gear can escape.

Wild Gears have a smooth surface over which the washers easily slide. My recollection from childhood is that Spirograph gears are not smooth so this may not work with them.

[This is true. Spirograph gears are die-cast and have lines and numbers on them. – Ed.]

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I was thinking some more about storage for the larger gears and rings and had a brainstorm. Here is the result.

I considered making something similar but this was less than $8 on Amazon and I figured even my time in retirement is worth more than that, so I bought it almost fully assembled. There are two minor issues. The large rings don’t stand properly in the back. The posts come up inside the ring. I had to put my largest gear as the rearmost item to give them something to lean against. Putting a heavy duty paper plate or maybe a light plastic one in the back would accomplish the same thing and is probably a better solution since it can be left there permanently.

The other issue is the smallest gear that fits properly is about 100 teeth. Anything smaller than that doesn’t rest on the side rails, and anything much smaller will fall right through. The rack ships as four pieces, with the cross dowels that hold it together needing to be fitted. As such, they could be shortened, pulling the side rails closer together and allowing smaller gears to be stored. I may cut an inch or so off of them and see how many more gears it will hold then.

I just now got this rack and haven’t used it while drawing yet, so I can’t say how well it will work in practice. I do know you don’t want to just cram in as many gears as will fit between the posts. There needs to be room to separate them so you can flip to the one you want. On the other hand, it wouldn’t be terrible to just lift out the whole group, find the one you want, and put the rest back. That would still be a lot faster than flipping through the CD case or digging through a box. Ten gears will fit between each set of posts.

Here is the bamboo dish rack on Amazon (affiliate link). They are inexpensive.

Anyone else try something like this? Let us know in the comments.

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More from Jay Heyl, who has provided some great ideas for Wild Gears storage. Jay has noticed that the laser cutting of the gears leaves an edge that is not perfectly smooth; it’s very slightly jagged, like steps on the X and Y axes, or like a pixelated edge if you blow up a digital image. I took out my jeweller’s loupe to look at some gears and I can see these lines, and I can feel their texture with a fine tool. You can kind of see it inside the holes and on the sides of the teeth in this photo:

Not satisfied with this edge, Jay writes:

I have one other rather geeky “modification” I’ve done to many of my Wild Gears…. smoothing out the stair step edges left by the laser when the gears are cut. These can cause an irregular movement as the pen slides around the hole and result in the lines having jagged sections. The stair step on the gear teeth can make the gears not slide against each other as they should. This can make the larger gears stick when using the outer holes and can make the smaller gears jump around. The stair steps eventually wear down from use but it can take a very long time for them to become anywhere near smooth.

Enter the diamond coated burr bit set. This set was less than $10 from Amazon.

This sloped conical burr (right) is a perfect fit for the small pen holes. You can do it totally by hand, simply rolling the bit between your fingers while sticking it in the pen hole, but that will likely wear your hands out pretty fast. I put the bit in a cordless drill and set it to screwdriver mode to slow the bit speed. Run the bit into the hole all the way to the end of the diamond section. Just a quick in and out is sufficient. I do all the holes from one side of the gear and then turn it over and re-do them from the other side.

You do need to watch what you’re doing when using the drill. These bits are covered in diamonds, the hardest natural substance known to man, and acrylic is not remotely close to the hardest substance known to man. With a short span of inattention you can turn a nice round pen hole into something more like a pen slot. This bit isn’t bad because the conical shape helps it center itself in the hole and it’s almost the perfect diameter. Quick in, quick out, you’re good to go.

None of the bits in this set is a perfect fit for the larger pen holes. I use one of the barrel bits and slide the gear around the bit when it’s inserted in the pen hole. I try to keep the gear moving so I don’t get the hole out of round. A couple moderately slow passes around the hole will knock the high points off the stair step and make the pen feel much smoother. You don’t need to get it perfectly smooth. Just knock down the edges and the pen won’t jump as it moves around the hole. You do need to be careful about keeping the gear perpendicular to the bit or you can take material off the rim of the hole and leave the ridges in the middle.

Smoothing the gear teeth takes two passes, the first with one of the medium barrel bits to deal with the outer portion of the teeth and then another with one of the smaller barrel bits that can fit down further between the teeth. The thing here is to watch that the gear is perpendicular to the bit so it takes a bit of material from the entire width of the teeth.

You don’t have to smooth both the gears and the rings. As long as the stair steps have been removed from one, the teeth will slide together smoothly. Imagine you had two staircases, one turned upside down on top of the other. They obviously won’t slide against each other very well. Replace one staircase with a sheet of plywood and the plywood will slide along the edges of the stairs. Same thing with the gears and rings. Doing both should make it a bit smoother but I haven’t found it necessary.

With all of these it’s probably a good idea to start doing it totally by hand until you have a feel for it and some confidence that you aren’t going to mess things up. Do a few holes and then try them with a pen. I think you’ll notice the difference, particularly if your gears are brand new. Ironically, the burr bits will probably leave a slight burr around the edges of the holes and gear teeth. Put the gear flat on a piece of paper and slide it around for ten or fifteen seconds. That will remove the burrs and make the gear feel smooth when drawing. ​

So if you’re seeking perfection, you might want to try what Jay’s been doing. The diamond burr sets even come in different grits, so a person could totally geek out getting the edges smoother and smoother. Or find out for yourself where the Law of Diminishing Returns applies. Feel free to share in the comments if you try this.

More from Jay Heyl coming soon.

The post For Wild Gears Super Geeks: Modifications appeared first on SpiroGraphicArt.

Watercolour paper stretches when wet, so needs to be taped down on a flat surface like a board. A better tape than what I’m using (beige masking tape or green painter’s tape) is gummed paper tape. You can buy artist’s tape or, apparently, the tape used by butchers. I need to get some, and perfect my stretching skills.

There are many other watercolour techniques to explore for creating beautiful backgrounds. Just do a search on YouTube.

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Meanwhile, here are some videos of pieces that I’ve done:

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Unlike Spirograph pieces, which have a rim supporting the toothed edges, Wild Gears are laser-cut from a flat sheet of acrylic, so using putty (as you can see in my videos) actually raises the gear a bit off the surface of the paper. Holding the gear down with very strong rare earth magnets instead would keep them in closer contact with the paper.

These rare earth magnets have screw-in hooks in the top. That’s handy, as you need something to grasp in order to move/remove them – they’re so strong.

Suze writes, “I leave the screw part of them off where it would interfere with my arm movement. The acrylic does interfere with the magnetic force so sometimes I put the magnets on an angle half on and half off the acrylic. I try to be gentle with these in case I damage the teeth of the ring.”

She notes that the white/magnetic boards that are sold in the budget shops are not strong enough, so she went for a commercial grade board.

The working surface of her board is 50 x 50 cm (almost 20 x 20 inches), so it accommodates the large Wild Gears frames, which are 15 x 15 inches (40 x 40 cm).

She can even use the whiteboard (with, I assume, a dry erase marker) to make larger drawings. If she tries to use it vertically, the moving gears fall out, but it works on an angle.

This would be a good setup if one were demonstrating to a group.

She find that she saves time setting up and putting away the Wild Gears, as she doesn’t have to mess with putty.

With enough magnets, the gears and the paper do not shift. She bought 14 magnets altogether. I looked on Amazon and these are the closest I found to the ones in her photo. They’re 16 mm (about 5/8″) in diameter. There are others that are larger.

She got her commercial quality magnetic board made by ABP Group (http://www.abpgroup.net.au/) in Brisbane. They’re made for schools and work environments. It might be something one would have to source locally. If you know of any sources, please mention them in the comments.

Thanks, Suze for your interesting contribution!

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